Solving By Elimination
Instead of graphing or substituting, you could also eliminate a variable entirely from a systems of equations. This is the elimination method. The elimination method is when opposite coefficients are added together to getzero of one variable. To eliminate, you have to either add or subtract. When eliminating, you must know what to eliminate and when to subtract or add.
When the corresponding coefficients in both equations are matching, you subtract. When subtracting for elimination, you are subtracting expressions that are equal from both sides to keep the equation balanced. Also when you find one variable, substitute it back in to find the other variable.
3x + y = 1
3x + 2y = 4
3x + y = 1
-(3x + 2y) = -4
-1y=-3
y=3
3x + 3 = 1
3x + 3 - 3 = 1 - 3
3x = -2
3x/3 = -2/3
x = -2/3
Adding works just as well. When you add, the coefficients need to be the same, except the signs must be different. When solving using addition, the steps are the same as subtracting, but you are adding so one variable comes out to zero.
3x + 2y = 4
+(x - 2y) +4
4x = 8
4x/4 = 8/4
x = 2
2 - 2y = 4
-2y = 2
y = -1
The elimination method works well when the coefficients are matching, but when they aren't, you must make the coefficients match. You do this by multiplying a number to one of the equations (the whole equation) so the coefficient matches with its correspondent in the other equation.
2x + 3y = -1
4x + 5y = -1
2(2x + 3y) = 2(-1)
4x + 6y = -2
4x + 6y = -2
-(4x + 5y) - (-1)
1y = -1
y = -1
2x + 3(-1) = -1
2x - 3 = -1
2x = 2
x = 1
When both variables get eliminated, there is no solution for the systems of equations. If it comes out to an always true statement, the systems of equations has infinite solutions.